Research on Dynamical Systems, Chaos and Fractals Modeled by Billiard Systems in Aspect of Global Analysis

改进了二维无食离散型台球的Lipschitz连续不变叶的存在性证明，使其更具建设性和初等性质。这可能会在应用方面取得新的进展。我们引入了Busemann的测地线几何来研究其位形空间中的台球轨迹。特别是，我们发现了平行线和周期轨迹之间的一些关系。此外，我们还用计算机模拟观察到了台球系统在|x/a|^r+|y/b|^r=1(r&gt；1)给出的闭合曲线上的有趣现象。它的Poincare映射提供了大量的信息，可以用来证明这些现象。在被认为是经典周期台球的玩具模型的双曲和间歇映射中，传输被发现是由Frobenius-Perron算子的谱来表征的。此外，我们还研究了边界元方法与量子台球散射问题的关系，研究了复杂动力学中不同周期点的重整化问题。我们定义了一个新的映射空间，它在抛物重整化及其扰动下是不变的。我们观察到可数到一个分段可逆马氏系统的大偏差。特别地，我们在一定条件下证明了(水平2)大偏差上界和指数递减性质。此外，我们还研究了大偏差定律的多重分形化，并成功地证明了Boyle和Maass的猜想。我们可以构造一个单参数的复杂结构族，其临界点为常返点，瞬变切换到另一个临界点。我们表明，在该点保持复发的变形自由度相对较低。作为量子混沌等相关课题的一部分，我们一直在研究算子代数理论本身，并取得了一些成果。

We have improved the proof of the existence of Lipschitz continuous invariant foliations for two dimensional dispersing billiards without eclipse and make it more constructive and more elementary. This may give a new progress in applications. We introduced the geometry of geodesies due to Busemann to study the billiard ball trajectories in their configuration spaces. In particular, we found out some relations between parallels and periodic trajectories. Moreover, we observed interesting phenomena of billiard systems in a closed curve given by |x/a|^r+|y/b|^r=1(r>1) with computer simulations. Its Poincare map gives a lot of information, by which those phenomena may be proved.In hyperbolic and intermittent maps regarded as toy models of classical periodic billiards, transports are found to be characterized by the spectra of the Frobenius-Perron operator. Furthermore, we researched the relationship between boundary elements method and scattering problems in quantum billiards.We studied the renormalization associated to indifferent periodic points of complex dynamics. We were able to define a new space of maps which is invariant under parabolic renormalization and its perturbations.We observed large deviations for countable to one piecewise invertible Markov systems. In particular, we showed the(level 2)upper large deviation bounds and exponential decreasing property under certain conditions. Moreover, we researched multifractal version of large deviation laws.Further, we have succeeded to prove the conjecture by Boyle and Maass. We can construct a one-parameter family of complex structures with critical point at the point the recurrence and the transience switch to the other. We showed that freedom of deformation to preserve recurrence at the point is relatively low. As related topics to Quantum Chaos and so on, we have been studying the theory of operator algebras itself and obtained several results in the subject.